An increasing variables singular system of fractional q-differential equations via numerical calculations

Abstract

We investigate the existence of solutions for an increasing variables singular m-dimensional system of fractional q-differential equations on a time scale. In this singular system, the first equation has two variables and the number of variables increases permanently. By using some fixed point results, we study the singular system under some different conditions. Also, we provide two examples involving practical algorithms, numerical tables, and some figures to illustrate our main results.

Iterative Positive Solutions to a Coupled Riemann-Liouville Fractional q -Difference System with the Caputo Fractional q -Derivative Boundary Conditions

“…Quantum calculus is not only used in quantum mechanics but also plays an important role in economics [20], dynamic system and quantum model [21], heat and wave equation [22], sampling signal analysis theory [23], and so on. Because of its wide application, fractional q-difference equation has entered the field of view of researchers [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”

“…At present, there are many theoretical methods to solve the boundary value problem of fractional q-difference equations. The majority use all kinds of the fixed point theorem methods, such as the Guo-Krasnoselskii's fixed point theorem [25][26][27][28][29], the Leray-Schauder alternative principle, and the Banach contraction mapping principle [30][31][32][33][34], and less use monotone iteration techniques [35][36][37][38][39]. All these methods can effectively study the existence of solutions, but the reason why we chose to use the monotone iterative method is that it has more advantages than other methods, which can not only prove the existence of positive solutions but also obtain numerical approximate solutions within certain limits of error.…”